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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004 
  
3D rotation can be expressed by a rotation angle, 0, and an axis 
of rotation given by unit vector components (€. &,, &) in the 
ECF frame. The sign and rotation angle follows the right-hand 
rule. Finally the quaternion (ql, q2, q3, q4) is related to 0 and 
(&, &y. &) by: 
q1 = &, sin(0/2) 
q2 = &, sin(6/2) 
q3 = €, sin(0/2)........ (2) 
q4 = cos(0/2) 
The image lines in the Level 1B product are sampled at a 
constant rate. This means that the imaging time can be 
computed directly from the given data avgLineRate (Average 
Line Rate) and firstLineTime (First Line Time) with no 
approximations: 
t=r/ avgLineRate * firstLineTime .... (3) 
One point on the imaging ray is the perspective centre of the 
virtual camera at time t. The coordinates of the perspective 
centre in the spacecraft coordinate system are constant and 
given data. In matrix notation: 
Cs (CC CL (4) 
Where Cx, Cy and C; are values in the camera calibration file 
(*.geo file). It is possible to locate the origin of the spacecraft 
coordinate system in the ECF system at a time t by interpolating 
the position time series in the ephemeris file. Let us call this 
position Sg(t). Likewise, we can find the attitude of the 
spacecraft coordinate system at a time (t) in the ECF system by 
interpolating the quaternion time series in the attitude file. This 
quaternion, q°g(t), represents the rotation from the ECF system 
to the spacecraft body system at time t. Then using quaternion 
algebra, the position of the perspective centre at time t in the 
ECF coordinate system is: 
Cet) = (@"s(1)" Cs q"s(0) + Se(0, or 
C(t) = q*&(t) Cs (q9r(t))" + Sp() (5) 
Alternatively, computing R(t), the rotation matrix from the 
given quaternions q'r(t) for time (t) as a rotation from the 
spacecraft body to the ECF, then (5) above can be expressed by: 
Cat) +R) Es + Spb oo (6) 
Expressions (5) and (6) are the position of the projection centre 
Xp =0 
Yp = -c*detPitch, with detPitch being the 
distance (in mm) between centres of adjacent pixels in the array 
To convert these detector coordinates to camera coordinates, it 
is necessary to apply the rotation and translation given by the 
following equations: 
Xc = cos(detRotAngle)* X - sin(detRotAngle)* Y, + 
detOriginX 
Xe = sin(detRotAngle)* Xp + cos(detRotAngle)* Y, + 
detOrieiny =... 0 (7) 
Zp = C (Virtual Principal Distance) 
With detRotAngle being the Rotation of the detector coordinate 
system as measured in the camera coordinates system in 
degrees. detOriginX and detOriginY are the X and Y 
coordinates of the pixel 0 in the linear detector array, in the 
camera coordinates system, in mm. detRotAngle, detOriginX 
and detOriginY are included in the calibration data file (*.geo). 
As Level IB images do not have lens distortion the corrected 
image point is identical to the measured image point, hence: 
Xc' 7 Xc 
Ye Em Yc Gat ahi we is ewe (8) 
Zo’ = Zc 
The unit vector wc that is parallel to the external ray in the 
camera coordinate system is just the position of (Xc', Yo, Zc^) 
relative to the perspective centre at (0, 0, 0), normalized by its 
length. In matrix notation, this vector is: 
We 7 (Xc', Ye Ze)" and wc 7 Wc/| Wei] (9) 
It is possible to convert this vector first to the spacecraft 
coordinate system and then to the ECF system. The unit 
quaternion for the attitude of the camera coordinate system, i.e., 
the quaternion for the rotation of spacecraft frame into the 
camera frame q^, is in the geometric calibration file (*.geo). 
Then, using quaternion algebra 
We = q's(0 qc Wc q'c q'«(t) or 
Wr 7 q^ k(t) qs we (9° (1) 4%)” or using matrix algebra, 
Wi = RED Rew, Vi plums Ww (10) 
The resulting multiplication matrix R"«(t) has following form: 
With: 
  
[R «(0] -(R'S()R* )'- 
? » 2 5 
ww + an — bu — ‘Cum 
2aubu) — 2w€u) 
  
Î 
2 2 2 2 
OQ t ao t bo tco 
2boco + 2wbu 
  
au bu) +; 20@wWCw 
2 2 2 2 
Q^ —aun + bw“ — Cm 
Du Cu) - 20a 
2xc - 20b 
2buco *- 2000 (11) 
2 2 2 2 
œ — au’ — bu)” +E€u) 
  
at the instant (t) expressed in the ECF coordinate system that 
May corresponds to a position of a GCP in the image. In this 
way it can be replaced in expressions (1) above for the position 
(line j) that corresponds to a time (t). 
For any column and row measurement (c, r) of a pixel in the 
image, the corresponding position of the image point in the 
detector coordinate system (relative to the centre of the lowest 
numbered pixel in the detector) is: 
697 
: constant term, being a function of both scalars qs (4) and 
s 
q c (4) : 
ag; by; cq: Are elements of the instantaneous rotation matrix 
[RF(t)]" also function of the quaternions q(t) for the instant 
(t) and q 
The above instantaneous matrix rotation can be used in 
expressions (1) above for the corresponding line (time) image 
where a CGP or an interest point is to be observed.